Question

ln is tangent to circle o at point m and qm is a diameter. determine the measure of the following angles. the measure of ∠qml is degrees. the measure of ∠pmn is degrees.

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. Since \(LN\) is tangent to circle \(O\) at \(M\) and \(OM\) is a radius and \(QM\) is a diameter (so \(O\) lies on \(QM\)), \(\angle OML = 90^{\circ}\).

Step2: Find \(\angle QML\)

We know that \(\angle QMR = 42^{\circ}\) and \(\angle PMQ\) and \(\angle QMR\) subtend the same arc \(QR\). The inscribed - angle theorem states that an inscribed angle is half of the central angle subtending the same arc. But we can also use the fact that \(\angle QML\) and \(\angle QMR\) are related as follows: \(\angle QML=90^{\circ}-\angle QMR\). Given \(\angle QMR = 42^{\circ}\), then \(\angle QML = 90 - 42=48^{\circ}\).

Step3: Find \(\angle PMN\)

We know that \(\angle PMQ\) and \(\angle QMR\) subtend the same arc \(QR\), so \(\angle PMQ=\angle QMR = 42^{\circ}\). Also, \(\angle OML = 90^{\circ}\). \(\angle PMN=90^{\circ}-\angle PMQ\). Since \(\angle PMQ = 27^{\circ}\), then \(\angle PMN=90 - 27 = 63^{\circ}\).

Answer:

The measure of \(\angle QML\) is \(48\) degrees.
The measure of \(\angle PMN\) is \(63\) degrees.